with \(\mu \)  and \(\nu \)  denoting the Lorentz indices of the incoming photon and outgoing \(\Lambda (1690)\) , respectively. \(M^{\nu \mu }_s\)  represents the \(s\) -channel amplitude of the nucleon and nucleon resonance exchanges, and \(M^{\nu \mu }_t\)  is the \(t\) -channel amplitude consisting of the \(K\)  and \(K^\ast \)  exchanges. They can be calculated straightforwardly with the effective Lagrangians, resonance propagators, and form factors given in the following part of this section. \(M^{\nu \mu }_{\rm int}\)  in Eq. (REF ) represents the generalized interaction current arising from the photon attaching to the internal structure of the \(\Lambda (1690)NK\)  vertex. It is known that in the effective Lagrangian approach, introducing the phenomenological form factors will result in violation of the gauge invariance of the reaction amplitudes. In the present work, we follow Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]} to introduce an auxiliary current \(C^\mu \)  to compensate the violation of the gauge invariance caused by the form factors and model the generalized interaction current \(M^{\nu \mu }_{\rm int}\)  as
\(M^{\nu \mu }_{\rm int} = \Gamma _{\Lambda ^\ast NK}^\nu (q) C^\mu + M^{\nu \mu }_{\rm KR} f_t, \) 
